url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | exact ⟨Line a b, Line_isLine h, by simp⟩ | case inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ ∃ L, L.IsLine ∧ {a, b} ⊆ L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ ∃ L, L.IsLine ∧ {a, b} ⊆ L
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ {a, b} ⊆ Line a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ {a, b} ⊆ Line a b
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : b ≠ a
⊢ {a, a} ⊆ Line a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : b ≠ a
⊢ {a, a} ⊆ Line a b
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | refine ⟨?_, ?_, ?_, hl⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ NotCollinear x y z | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ y
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ z
case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ y ≠ z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ NotCollinear x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ y | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ y
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x z | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
⊢ False | case refine_1.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, z} ⊆ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_1.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, z} ⊆ L
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, z} ⊆ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ z | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x y | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
⊢ False | case refine_2.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_2.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ y ≠ z | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ y ≠ z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x y | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
⊢ False | case refine_3.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_3.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.rotate | [110, 1] | [112, 54] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {v, w, u} ⊆ l
⊢ {u, v, w} ≤ {v, w, u} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {v, w, u} ⊆ l
⊢ {u, v, w} ≤ {v, w, u}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.swap | [114, 1] | [116, 54] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {w, v, u} ⊆ l
⊢ {u, v, w} ≤ {w, v, u} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {w, v, u} ⊆ l
⊢ {u, v, w} ≤ {w, v, u}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | let S : Set (Set V) := setOf Set.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
⊢ Set.univ.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
⊢ Set.univ.IsLine | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have : S.Nonempty := let ⟨x, y, hxy⟩ := exists_pair_ne V; ⟨_, Line_isLine hxy⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
⊢ Set.univ.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
⊢ Set.univ.IsLine | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨L, hL, hL'⟩ := S.toFinite.exists_maximal_wrt id S this | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, id L ≤ id a' → id L = id a'
⊢ Set.univ.IsLine | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | dsimp at hL' | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, id L ≤ id a' → id L = id a'
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ Set.univ.IsLine | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, id L ≤ id a' → id L = id a'
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | suffices L = Set.univ by rwa [← this] | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ L = Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rw [Set.eq_univ_iff_forall] | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ L = Set.univ | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ ∀ (x : V), x ∈ L | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ L = Set.univ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | by_contra! | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ ∀ (x : V), x ∈ L | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : ∃ x, x ∉ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ ∀ (x : V), x ∈ L
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨a, b, hab, rfl⟩ := hL | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : ∃ x, x ∉ L
⊢ False | case intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
this : ∃ x, x ∉ Line a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : ∃ x, x ∉ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨c, hc'⟩ := this | case intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
this : ∃ x, x ∉ Line a b
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
this : ∃ x, x ∉ Line a b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have hac : a ≠ c := fun h => hc' (subset_generateLine _ (by simp [h])) | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have hbc : b ≠ c := fun h => hc' (subset_generateLine _ (by simp [h])) | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp only [NotCollinear, not_and, not_forall, not_not] at h | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨M, hM, habc⟩ := h a b c hab hac hbc | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have := hL' M hM (Set.IsLine.generateLine_subset (habc.trans' (by simp)) hM) | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rw [this] at hc' | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | exact hc' (habc (by simp)) | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rwa [← this] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : L = Set.univ
⊢ Set.univ.IsLine | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : L = Set.univ
⊢ Set.univ.IsLine
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp [h] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
h : a = c
⊢ c ∈ {a, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
h : a = c
⊢ c ∈ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp [h] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
h : b = c
⊢ c ∈ {a, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
h : b = c
⊢ c ∈ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ {a, b} ≤ {a, b, c} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ {a, b} ≤ {a, b, c}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ c ∈ {a, b, c} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ c ∈ {a, b, c}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | let S : Set (V × V × V) := setOf (fun ⟨a, b, c⟩ => NotCollinear a b c) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : S.Nonempty := let ⟨x, y, z, hxyz⟩ := h; ⟨(x, y, z), hxyz⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | let f : V × V × V → ℝ := fun ⟨a, b, c⟩ => dist a b + dist b c + dist c a | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨⟨a, b, c⟩, (h₁ : NotCollinear _ _ _), h₂⟩ := S.toFinite.exists_minimal_wrt f S this | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ a' ∈ S, f a' ≤ f (a, b, c) → f (a, b, c) = f a'
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [Prod.forall, Set.mem_setOf_eq] at h₂ | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ a' ∈ S, f a' ≤ f (a, b, c) → f (a, b, c) = f a'
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ a' ∈ S, f a' ≤ f (a, b, c) → f (a, b, c) = f a'
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace h₂ : ∀ a' b' c' : V, NotCollinear a' b' c' →
dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' := by
intro a' b' c' hL
by_contra! h
exact h.ne' (h₂ a' b' c' hL h.le) | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleTriangle x y z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [SimpleTriangle] | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleEdges.Adj x y ∧ SimpleEdges.Adj y z ∧ SimpleEdges.Adj z x ∧ NotCollinear x y z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleTriangle x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | by_contra! cont | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleEdges.Adj x y ∧ SimpleEdges.Adj y z ∧ SimpleEdges.Adj z x ∧ NotCollinear x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleEdges.Adj x y ∧ SimpleEdges.Adj y z ∧ SimpleEdges.Adj z x ∧ NotCollinear x y z
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | wlog hab : ¬ SimpleEdges.Adj a b generalizing a b c | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ False | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : ¬¬SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ¬SimpleEdges.Adj a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [SimpleEdges_adj, h₁.1, ne_eq, not_false_eq_true, true_and, not_forall, not_not] at hab | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ¬SimpleEdges.Adj a b
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ∃ x, sbtw a x b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ¬SimpleEdges.Adj a b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨d, adb⟩ := hab | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ∃ x, sbtw a x b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ∃ x, sbtw a x b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have habc : c ∉ Line a b := fun hc ↦ h₁.2.2.2 (Line a b) (Line_isLine h₁.1)
(by simp [left_mem_Line, right_mem_Line, hc]) | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have hdab : d ∈ Line a b := middle_extend_mem_Line adb | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : dist d c < dist d b + dist b c := by
by_contra!
refine habc (generateLine_close_right hdab right_mem_Line ?_)
exact ⟨adb.ne23, hcd.symm, h₁.2.2.1, this.antisymm (dist_triangle _ _ _)⟩ | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a := by
linarith only [this, adb.2.2.2] | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace : ¬ NotCollinear a d c := fun h => (h₂ a d c h).not_lt this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ¬NotCollinear a d c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [notCollinear_iff, adb.ne12, hcd.symm, h₁.2.1, true_and, not_and, forall_true_left,
ne_eq, not_forall, not_not, exists_prop, not_false_eq_true] at this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ¬NotCollinear a d c
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ∃ x, x.IsLine ∧ {a, d, c} ⊆ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ¬NotCollinear a d c
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨L, hL, hL'⟩ := this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ∃ x, x.IsLine ∧ {a, d, c} ⊆ x
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : {a, d, c} ⊆ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ∃ x, x.IsLine ∧ {a, d, c} ⊆ x
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [Set.subset_def, Set.mem_singleton_iff, Set.mem_insert_iff, forall_eq_or_imp,
forall_eq] at hL' | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : {a, d, c} ⊆ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : {a, d, c} ⊆ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : b ∈ L := hL.close_right hL'.1 hL'.2.1 adb | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | refine h₁.2.2.2 L hL ?_ | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ {a, b, c} ⊆ L | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp [this, hL'.1, hL'.2.2] | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ {a, b, c} ⊆ L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ {a, b, c} ⊆ L
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | intro a' b' c' hL | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
⊢ dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | by_contra! h | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
⊢ dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
h : dist a' b' + dist b' c' + dist c' a' < dist a b + dist b c + dist c a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
⊢ dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact h.ne' (h₂ a' b' c' hL h.le) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
h : dist a' b' + dist b' c' + dist c' a' < dist a b + dist b c + dist c a
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
h : dist a' b' + dist b' c' + dist c' a' < dist a b + dist b c + dist c a
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | rw [not_not] at hab | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : ¬¬SimpleEdges.Adj a b
⊢ False | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ (a b c : V) →
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
MetricSpace V | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : ¬¬SimpleEdges.Adj a b
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | refine cont a b c hab ?_ ?_ h₁ | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ (a b c : V) →
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
MetricSpace V | case intro.mk.mk.intro.inr.refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj b c
case intro.mk.mk.intro.inr.refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj c a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ (a b c : V) →
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
MetricSpace V
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact not_not.1 <| this b c a h₁.rotate <|
fun _ _ _ h => (h₂ _ _ _ h).trans_eq' <| by ring | case intro.mk.mk.intro.inr.refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro.inr.refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj b c
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | ring | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
x✝² x✝¹ x✝ : V
h : NotCollinear x✝² x✝¹ x✝
⊢ dist b c + dist c a + dist a b = dist a b + dist b c + dist c a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
x✝² x✝¹ x✝ : V
h : NotCollinear x✝² x✝¹ x✝
⊢ dist b c + dist c a + dist a b = dist a b + dist b c + dist c a
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact not_not.1 <| this c a b h₁.rotate.rotate <|
fun _ _ _ h => (h₂ _ _ _ h).trans_eq' <| by ring | case intro.mk.mk.intro.inr.refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj c a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.mk.intro.inr.refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj c a
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | ring | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
x✝² x✝¹ x✝ : V
h : NotCollinear x✝² x✝¹ x✝
⊢ dist c a + dist a b + dist b c = dist a b + dist b c + dist c a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
x✝² x✝¹ x✝ : V
h : NotCollinear x✝² x✝¹ x✝
⊢ dist c a + dist a b + dist b c = dist a b + dist b c + dist c a
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp [left_mem_Line, right_mem_Line, hc] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
hc : c ∈ Line a b
⊢ {a, b, c} ⊆ Line a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
hc : c ∈ Line a b
⊢ {a, b, c} ⊆ Line a b
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | by_contra! | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
⊢ dist d c < dist d b + dist b c | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
⊢ dist d c < dist d b + dist b c
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | refine habc (generateLine_close_right hdab right_mem_Line ?_) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ sbtw d b c | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact ⟨adb.ne23, hcd.symm, h₁.2.2.1, this.antisymm (dist_triangle _ _ _)⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ sbtw d b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d b + dist b c ≤ dist d c
⊢ sbtw d b c
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | linarith only [this, adb.2.2.2] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ dist a d + dist d c + dist c a < dist a b + dist b c + dist c a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_comm | [198, 1] | [199, 41] | simp only [Delta, add_comm, dist_comm] | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
⊢ Delta u v w = Delta w v u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
⊢ Delta u v w = Delta w v u
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | rw [Delta] | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
⊢ 0 < Delta u v w | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
⊢ 0 < dist u v + dist v w - dist u w | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
⊢ 0 < Delta u v w
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | by_contra! | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
⊢ 0 < dist u v + dist v w - dist u w | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this : dist u v + dist v w - dist u w ≤ 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
⊢ 0 < dist u v + dist v w - dist u w
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | have : sbtw u v w := sbtw_mk h.1 h.2.2.1 (by linarith only [this]) | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this : dist u v + dist v w - dist u w ≤ 0
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this✝ : dist u v + dist v w - dist u w ≤ 0
this : sbtw u v w
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this : dist u v + dist v w - dist u w ≤ 0
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | exact h.2.2.2 (Line u v) (Line_isLine this.ne12) (by simp [right_extend_mem_Line this]) | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this✝ : dist u v + dist v w - dist u w ≤ 0
this : sbtw u v w
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this✝ : dist u v + dist v w - dist u w ≤ 0
this : sbtw u v w
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | linarith only [this] | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this : dist u v + dist v w - dist u w ≤ 0
⊢ dist u v + dist v w ≤ dist u w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this : dist u v + dist v w - dist u w ≤ 0
⊢ dist u v + dist v w ≤ dist u w
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Delta_pos_of | [201, 1] | [205, 90] | simp [right_extend_mem_Line this] | V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this✝ : dist u v + dist v w - dist u w ≤ 0
this : sbtw u v w
⊢ {u, v, w} ⊆ Line u v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u✝ v✝ w✝ : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z u v w : V
h : NotCollinear u v w
this✝ : dist u v + dist v w - dist u w ≤ 0
this : sbtw u v w
⊢ {u, v, w} ⊆ Line u v
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | by_contra! h' | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
⊢ ∃ c ∈ Line a b, sbtw c a b ∨ sbtw a c b ∨ sbtw a b c | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
⊢ ∃ c ∈ Line a b, sbtw c a b ∨ sbtw a c b ∨ sbtw a b c
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact h this | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
this : Line a b = {a, b}
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
this : Line a b = {a, b}
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | refine (subset_generateLine _).antisymm' ?_ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ Line a b = {a, b} | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ GenerateLine {a, b} ⊆ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ Line a b = {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | change Line a b ⊆ {a, b} | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ GenerateLine {a, b} ⊆ {a, b} | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ Line a b ⊆ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ GenerateLine {a, b} ⊆ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | refine generateLine_minimal le_rfl ?_ ?_ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ Line a b ⊆ {a, b} | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x y z → z ∈ {a, b}
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x z y → z ∈ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ Line a b ⊆ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | rintro x y z (rfl | rfl) (rfl | rfl) h | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x y z → z ∈ {a, b} | case refine_1.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y y z
⊢ z ∈ {y, b}
case refine_1.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x y z
⊢ z ∈ {x, y}
case refine_1.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x y z
⊢ z ∈ {y, x}
case refine_1.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y y z
⊢ z ∈ {a, y} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x y z → z ∈ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact (h.ne12 rfl).elim | case refine_1.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y y z
⊢ z ∈ {y, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y y z
⊢ z ∈ {y, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact ((h' z (right_extend_mem_Line h)).2.2 h).elim | case refine_1.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x y z
⊢ z ∈ {x, y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x y z
⊢ z ∈ {x, y}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact ((h' z (left_extend_mem_Line h.symm)).1 h.symm).elim | case refine_1.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x y z
⊢ z ∈ {y, x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x y z
⊢ z ∈ {y, x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact (h.ne12 rfl).elim | case refine_1.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y y z
⊢ z ∈ {a, y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y y z
⊢ z ∈ {a, y}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | rintro x y z (rfl | rfl) (rfl | rfl) h | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x z y → z ∈ {a, b} | case refine_2.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y z y
⊢ z ∈ {y, b}
case refine_2.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x z y
⊢ z ∈ {x, y}
case refine_2.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x z y
⊢ z ∈ {y, x}
case refine_2.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y z y
⊢ z ∈ {a, y} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
hab : a ≠ b
h : Line a b ≠ {a, b}
h' : ∀ c ∈ Line a b, ¬sbtw c a b ∧ ¬sbtw a c b ∧ ¬sbtw a b c
⊢ ∀ {x y z : V}, x ∈ {a, b} → y ∈ {a, b} → sbtw x z y → z ∈ {a, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact (h.ne13 rfl).elim | case refine_2.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y z y
⊢ z ∈ {y, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ b y z : V
hab : y ≠ b
h✝ : Line y b ≠ {y, b}
h' : ∀ c ∈ Line y b, ¬sbtw c y b ∧ ¬sbtw y c b ∧ ¬sbtw y b c
h : sbtw y z y
⊢ z ∈ {y, b}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact ((h' z (middle_extend_mem_Line h)).2.1 h).elim | case refine_2.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x z y
⊢ z ∈ {x, y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : x ≠ y
h✝ : Line x y ≠ {x, y}
h' : ∀ c ∈ Line x y, ¬sbtw c x y ∧ ¬sbtw x c y ∧ ¬sbtw x y c
h : sbtw x z y
⊢ z ∈ {x, y}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact ((h' z (middle_extend_mem_Line h.symm)).2.1 h.symm).elim | case refine_2.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x z y
⊢ z ∈ {y, x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x✝ y✝ z✝ x y z : V
hab : y ≠ x
h✝ : Line y x ≠ {y, x}
h' : ∀ c ∈ Line y x, ¬sbtw c y x ∧ ¬sbtw y c x ∧ ¬sbtw y x c
h : sbtw x z y
⊢ z ∈ {y, x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_third | [207, 1] | [224, 15] | exact (h.ne13 rfl).elim | case refine_2.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y z y
⊢ z ∈ {a, y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y✝ z✝ a y z : V
hab : a ≠ y
h✝ : Line a y ≠ {a, y}
h' : ∀ c ∈ Line a y, ¬sbtw c a y ∧ ¬sbtw a c y ∧ ¬sbtw a y c
h : sbtw y z y
⊢ z ∈ {a, y}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | intro h | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
⊢ ¬SimpleEdges.Adj b d | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
⊢ ¬SimpleEdges.Adj b d
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | have hbcd : NotCollinear b c d := by
refine ⟨habc.2.1.ne, h.ne, hcd.ne, fun l hl h ↦ ?_⟩
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, Set.subset_def, forall_eq_or_imp,
forall_eq] at h
refine habc.2.2.2.2.2.2 l hl (by simp [*, hl.close_left h.2.1 h.2.2 hacd]) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : NotCollinear b c d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | replace hbcd : SimpleTriangle b c d := ⟨habc.2.1, hcd, h.symm, hbcd⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : NotCollinear b c d
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : NotCollinear b c d
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | have habd : sbtw a b d := by
refine sbtw_mk habc.1.ne h.ne ?_
have := habc_min _ _ _ hbcd
rw [Delta, Delta] at this
linarith [hacd.dist] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | refine habc.2.2.2.2.2.2 (Line a d) (Line_isLine habd.ne13) ?_ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ {a, b, c} ⊆ Line a d | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | eqn_7 | [226, 1] | [243, 44] | simp [middle_extend_mem_Line, hacd, habd] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ {a, b, c} ⊆ Line a d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b c d : V
habc : SimpleTriangle a b c
habc_min : ∀ (a' b' c' : V), SimpleTriangle a' b' c' → Delta a b c ≤ Delta a' b' c'
hacd : sbtw a c d
hcd : SimpleEdges.Adj c d
h : SimpleEdges.Adj b d
hbcd : SimpleTriangle b c d
habd : sbtw a b d
⊢ {a, b, c} ⊆ Line a d
TACTIC:
|
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